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MindMap Gallery Vector algebra and spatial analytic geometry
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Vector algebra and spatial analytic geometry
Mind map of vector algebra and spatial analytic geometry. The content includes vectors and their linear operations, the product of two vectors, plane equations, space surfaces and their equations, and straight line equations. Let's learn together.
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Vector algebra and spatial analytic geometry
- bacteria
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Vector algebra and spatial analytic geometry
Vectors and their linear operations
The concept of vector: vector
The unit vector and its modulus are 1 (i, j, k are the direction vectors on the x, y, and z axes respectively)
Positional relationship of vectors
Vertical a·b=|a|·|b|·cos<a·b>=0
Parallel (collinear)
Corresponding coordinates are proportional to
a=enter b
vector product of vectors
a·b=|a|·|b|·cos<a·b>
a·b=multiply and add the corresponding coordinates
Linear operations
Triangle Rule (connect end to end, connect end to end)
parallelogram rule
Commutative law of addition, associative law, associative law of multiplication, distributive law
The result of dividing a nonzero vector by its modulus is a unit vector in the same direction as the original vector.
Space rectangular coordinate system (x, y, z)
The first hexagram, the second hexagram, the third hexagram, the fourth hexagram
The coordinates of the point on the vector
midpoint coordinates between two points
The angles α, β, and γ between the vector and the x, y, and z axes respectively, (cosα)² (cosβ)² (cosγ)²=1
projection
The projection of b on a = b·cosθ
0<Ψ<90°, the projection is positive
Ψ=90°, the projection is zero
90°<Ψ<180°, the projection is negative
product of two vector quantities
Concept: a丄c, c丄b→c=|a×b|=|a|·|b|·sin<a·b>→=1/2 area of triangle = area of parallelogram
a×b=-b×a
Coordinate operations: third-order determinant
plane equation
Point French equation
plane general equation
Passing the origin: D=0
Parallel to the coordinate axis
Parallel to X axis→A=0
Parallel to Y axis→B=0
Parallel to Z axis→C=0
plane through coordinate axis
X-axis: A=D=0
Y axis: B=D=0
Z axis: C=D=0
Parallel to the xoy plane
parallel to x-axis
Parallel to the y-axis
intercept equation
Substitute the point and eliminate the same element → the general plane equation is transformed into an intercept equation
Space surfaces and their equations
Spherical equation
three forms
Surface of revolution
Rotation of a line (straight line or curve) about an axis
Whoever rotates around remains unchanged (changed from the original equation)
cylinder
y²=2x, the busbar is parallel to the z-axis → parabolic cylinder
elliptical cylinder
hyperbolic cylinder
Which axis is the bus parallel to and which axis does not appear?
general equations of space curves
Surface 1
Surface 2
intersection equation
parametric equations
parametric equations of a circle
x=r·cosθ
y=r·sinθ
Write z from x, y
Equation of a straight line
Point-wise equation
point
Direction vector: parallel to the original vector
two point equation
two o'clock
The denominator of the equation is zero, and the numerator of the corresponding term is also understood to be zero.
If the denominator of the x term is zero, it is understood that the plane passes through the x-axis
parametric equations
Let the two-point equation = t, and sort out the x, y, and z axes containing t
Find the intersection point of the line and the plane, substitute it into the plane equation, and find the t value
general equation
Intersection of two planes → solution of two plane equations
Positional relationship
line line
cosθ
Line surface
Normal vectors of lines and surfaces
Find the projection of a straight line on a plane
①Plane beam passing through a straight line
② Assuming that the plane 丄 is known, the normal vectors of the two planes are also perpendicular to each other, so the entry and exit can be found
③The intersection of the plane and the known plane is the projection of the straight line
plane beam equation
Use vertical relationships to find vertical surfaces
Projection of a point on a plane
①The normal vector of the plane is the direction vector of the point, thus writing the point-direction equation of the straight line
② Find the intersection point of the straight line and the plane, using parametric equations
distance from point to straight line
Find the point-direction equation of the straight line and substitute it into the formula